
We obtain the tree decomposition of the graph \(G\)  in \(\mbox{{\sf L}}\)  via [1]}.

Using Lemm REF , we can obtain the PEO of \(G\)  and also construct
the Elimination tree \(T(G,o)\)  in \(\mbox{{\sf L}}\) .

From \(T(G,o)\)  and \(G\) , we can obtain \(m\)  and \(r\)  and subsequently,
each element of \(N^+(G,o,i)\)  and \(N^-(T(G,o), o, i)\)  in \(\mbox{{\sf L}}\) .

We can compute the \((k+1)\) -coloring of \(G\) , \(col : V_G \rightarrow [k+1]\)  in \(\mbox{{\sf L}}\)  via [1]}
(Proof of Lemma 4.1).

